A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating…
A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_{c}$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is
more than half-filled if $\rho_{c}$ is less than $0.5$.
more than half-filled if $\rho_{c}$ is more than 1.0.
half-filled if $\rho_{c}$ is more than $0.5$.
less than half-filled if $\rho_{c}$ is less than $0.5$.
Solution
Let $V_{1}=$ total material volume of cylindrical shell $V_{2}=$ total inside volume of cylindrical shell and $y=$ fraction of $V_{2}$ volume filled with water.
Total weight $=$ Upthrust (Condition of floatation)
$\therefore \quad V_{1} \rho_{c} g+\left(y V_{2}\right)(1) g=\left(\frac{V_{1}+V_{2}}{2}\right)(1) g$
or $y=0.5+\left(0.5-\rho_{c}\right) \frac{V_{1}}{V_{2}}$
So, if $\rho_{c} < 0.5$ then $y>0.5$